I am studying client's request arrival patterns on web and application servers. About web server's request arrival pattern I read that "The request arrival rate on web server follows Poisson distribution". And about application server I read the sentence "The request arrival rate on application server follows exponential distribution. Now kindly explain " is there any difference between Poisson and exponential distribution in the context of client's request arrival pattern on server". Is there any difference between poisson and exponential distributions?
[Math] difference between poisson and exponential distributions in the context of client server systems
exponential functionpoisson distribution
Related Solutions
As A.S.'s comment indicates, both distributions relate to the same kind of process (a Poisson process), but they govern different aspects: The Poisson distribution governs how many events happen in a given period of time, and the exponential distribution governs how much time elapses between consecutive events.
By way of analogy, suppose that we have a different process, in which events occur exactly every $10$ seconds. Then the number of events that happen in a minute (i.e., $60$ seconds) is deterministically $6$, and the amount of time that elapses between consecutive events is, of course, deterministically $10$ seconds.
In contrast, in a Poisson process with a mean rate of one event every $10$ seconds (i.e., $\lambda = 1/10$), the number of events that happen in a minute is not deterministically $6$, but it has a mean of $6$. The exact distribution is given by the Poisson distribution:
$$ P_k(t) = \frac{(\lambda t)^k}{k!} e^{-\lambda t} $$
where $t = 60$ seconds is the time window. Thus, the probability that no events occur in a minute is given by
$$ P_0(t) = \frac{(6)^0}{0!} e^{-6} = e^{-6} \doteq 0.0024788 $$
whereas the probability that $6$ events occur in a minute is given by
$$ P_6(t) = \frac{(6)^6}{6!} e^{-6} = \frac{46656}{720} e^{-6} \doteq 0.16062 $$
That is obviously much more likely, as you would expect.
Similarly, the time between events is also not deterministically $10$ seconds, but it has a mean of $10$ seconds. The actual time distribution is the exponential distribution, which can be specified using its CDF (cumulative distribution function)
$$ F_T(t) \equiv P(T < t) = 1-e^{-\lambda t} $$
The CDF provides essentially the same information as the PDF (probability density function), whose formulation you gave in your question; in fact, the derivative of the CDF is the PDF. However, the CDF is sometimes easier to understand intuitively, so I'll explain using the CDF here.
In this case, the probability that the time between events is less than $10$ seconds is $F_T(10) = 1-e^{-1} \doteq 0.63212$, whereas the probability that the time between events is less than $60$ seconds is $F_T(60) = 1-e^{-6} \doteq 0.99752$. The probability that it is greater than $60$ seconds is $1-F_T(60) = e^{-6} \doteq 0.0024788$, and you'll notice this is equal to the probability that no events occur in a given minute. This is no coincidence; in a Poisson process, which is memoryless, the probability that the time between events is greater than a minute is naturally equal to the probability that no events occur in that minute!
The Poisson process can be interpreted as the process that counts the total number of events that have occurred when the waiting times between events are i.i.d. exponential.
Hence, if you find the holding/waiting times between some random events are i.i.d. and distributed exponentially with parameter $\lambda$, then the total number of events will be distributed as a Poisson distribution with parameter $\lambda t$.
Best Answer
The Poisson models the number of arrivals in a certain fixed time. It is a discrete distribution, taking on values $0,1,2,\dots$. The exponential models the waiting time between consecutive arrivals. It is a continuous distribution. There is a connection, since they are used in modelling two different features of the same phenomenon. But they are quite different distributions.
The connection is that if the waiting times between any two consecutive arrivals are independent exponentially distributed with parameter $\lambda$, then the number of events in unit time has Poisson distribution with parameter $\lambda$.